Subgroups — #32979099

Order 18688.

494 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988666	C₂	1	1	yes	maximal, characteristic
2	#22610059	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609932	C₂	1	2	yes	maximal
4	#22607815	C₂	1	2	yes	maximal, complemented
5	C₂×#334×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11924011	—	2	1	no	—
7	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
8	#17988666	—	2	2	no	—
9	#17988666	C₄	1	4	yes	complemented
10	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
11	C₂×#14556513	—	2	2	no	—
12	C₂×#14556503	—	2	1	no	—
13	#17988457	C₂²	1	2	yes	—
14	#17988350	—	2	2	no	—
15	#17988350	—	2	1	no	—
16	#17988350	—	2	1	no	—
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#334×C₇₃	—	2	2	no	—
20	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
21	C₂²×C₄×D₇₃	—	2	1	no	—
22	C₂×#11924011	—	2	4	no	—
23	C₂×#11924011	—	4	4	no	—
24	C₂²×#10596831	—	2	1	no	—
25	C₄×D₂₉₂	—	2	1	no	—
26	C₄×D₂₉₂	—	2	1	no	—
27	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
28	C₄×D₂₉₂	Q₈	1	2	yes	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄⋊Dic₇₃	—	2	2	no	—
32	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
33	C₂×#11924005	—	2	1	no	—
34	C₄⋊#10596831	—	2	1	no	—
35	C₄⋊#10596831	C₂×C₄	1	2	yes	—
36	#14556513	—	4	4	no	—
37	#14556503	—	2	4	no	—
38	C₂²×C₈×C₇₃	—	2	2	no	—
39	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₃	—	2	2	no	—
43	C₄⋊C₈×C₇₃	—	2	2	no	—
44	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₇₃	D₄	1	2	yes	—
46	C₂³×D₇₃	—	2	1	no	—
47	C₂×D₂₉₂	—	2	2	no	—
48	C₂×D₂₉₂	—	2	2	no	—
49	C₂×D₂₉₂	—	2	1	no	—
50	C₂×D₂₉₂	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₇₃	—	2	1	no	—
52	C₂²×Dic₇₃	—	4	2	no	—
53	C₂×C₄×D₇₃	—	2	2	no	—
54	C₂×C₄×D₇₃	—	2	2	no	—
55	C₂×C₄×D₇₃	—	4	2	no	—
56	C₂×C₄×D₇₃	—	4	2	no	—
57	#11924011	—	4	8	no	—
58	C₂×#10596831	—	2	2	no	—
59	C₂×#10596831	—	4	2	no	—
60	C₄⋊Dic₇₃	—	2	1	no	—
61	C₄⋊Dic₇₃	—	4	2	no	—
62	C₄⋊Dic₇₃	—	4	2	no	—
63	C₄⋊Dic₇₃	—	4	2	no	—
64	C₄⋊Dic₇₃	—	4	2	no	—
65	C₄⋊Dic₇₃	—	8	2	no	—
66	C₄⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
67	#11924005	—	2	2	no	—
68	#11924005	—	2	2	no	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
72	C₂×C₈×C₇₃	—	2	2	no	—
73	C₂×C₈×C₇₃	—	2	2	no	—
74	C₂×C₈×C₇₃	—	4	2	no	—
75	C₂×C₈×C₇₃	—	4	2	no	—
76	C₂×C₈×C₇₃	—	4	2	no	—
77	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₇₃	#41	1	1	yes	characteristic
83	C₂³×C₇₃	#121	1	1	yes	characteristic
84	C₂²×D₇₃	—	2	2	no	—
85	C₂²×D₇₃	—	4	2	no	—
86	C₂²×D₇₃	—	4	1	no	—
87	C₂²×D₇₃	—	4	1	no	—
88	C₂²×D₇₃	—	4	2	no	—
89	C₂×C₄×C₇₃	—	2	2	no	—
90	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	#135	1	2	yes	—
97	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
98	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
99	C₂×C₄×C₇₃	#137	1	2	yes	—
100	C₂×Dic₇₃	—	2	2	no	—
101	C₂×Dic₇₃	—	4	2	no	—
102	C₂×Dic₇₃	—	8	2	no	—
103	C₂×Dic₇₃	—	8	2	no	—
104	C₂×Dic₇₃	—	8	2	no	—
105	D₂₉₂	—	2	4	no	—
106	D₂₉₂	—	4	2	no	—
107	C₄×D₇₃	—	4	2	no	—
108	C₄×D₇₃	—	4	2	no	—
109	C₄×D₇₃	—	4	4	no	—
110	C₈×C₇₃	—	2	2	no	—
111	C₈×C₇₃	—	2	2	no	—
112	C₈×C₇₃	—	8	2	no	—
113	#10596831	—	4	2	no	—
114	C₂²×C₇₃	#380	1	1	yes	characteristic
115	C₂²×C₇₃	#463	1	2	yes	—
116	C₂²×C₇₃	#469	1	1	yes	characteristic
117	C₂²×C₇₃	#470	1	1	yes	characteristic
118	C₂²×C₇₃	#471	1	2	yes	—
119	C₂×D₇₃	—	4	2	no	—
120	C₂×D₇₃	—	4	2	no	—
121	C₂×D₇₃	—	4	2	no	—
122	C₂×D₇₃	—	8	2	no	—
123	C₂×D₇₃	—	8	2	no	—
124	C₄×C₇₃	—	2	1	no	—
125	C₄×C₇₃	—	2	1	no	—
126	C₄×C₇₃	#416	1	1	yes	characteristic
127	C₄×C₇₃	#418	1	1	yes	characteristic
128	C₄×C₇₃	#428	1	2	yes	—
129	C₄×C₇₃	#468	1	2	yes	—
130	C₄×C₇₃	#478	1	2	yes	—
131	Dic₇₃	—	4	2	no	—
132	Dic₇₃	—	16	2	no	—
133	#8656	—	73	1	no	maximal, Sylow
134	C₂×C₇₃	#1701	1	2	yes	—
135	C₂×C₇₃	#1703	1	2	yes	—
136	C₂×C₇₃	#1804	1	1	yes	characteristic
137	C₂×C₇₃	#1809	1	1	yes	characteristic
138	C₂×C₇₃	#1815	1	1	yes	characteristic
139	D₇₃	—	8	2	no	—
140	C₂×#436	—	73	1	no	—
141	#1834	—	73	1	no	—
142	#1802	—	73	2	no	—
143	C₂×#334	D₇₃	1	1	yes	characteristic, complemented
144	#1296	—	73	2	no	—
145	C₇₃	#8656	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	146	1	no	—
147	C₂×C₄×D₄	—	73	1	no	—
148	#436	—	73	4	no	—
149	#436	—	146	2	no	—
150	C₂×C₄⋊C₈	—	73	1	no	—
151	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
152	C₂×#102	—	146	2	no	—
153	C₂×C₂²⋊C₈	—	146	1	no	—
154	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
155	#384	—	73	2	no	—
156	#340	—	146	2	no	—
157	#340	—	146	1	no	—
158	#340	—	146	1	no	—
159	#334	—	2	2	no	—
160	C₂²×D₄	—	73	1	no	—
161	C₂³×C₄	—	146	1	no	—
162	C₂×M₁₆	—	292	4	no	—
163	C₂×M₁₆	—	146	4	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	146	1	no	—
166	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
167	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
168	C₄×D₄	—	73	2	no	—
169	C₄×D₄	—	73	2	no	—
170	C₄×D₄	—	146	1	no	—
171	C₄×D₄	—	146	1	no	—
172	C₂×C₄⋊C₄	—	73	1	no	—
173	C₂×C₄⋊C₄	—	146	2	no	—
174	C₂×C₄⋊C₄	—	146	2	no	—
175	C₂×C₄⋊C₄	—	146	2	no	—
176	C₂×C₂²⋊C₄	—	146	1	no	—
177	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	73	2	no	—
181	C₄⋊C₈	—	146	1	no	—
182	#102	—	292	4	no	—
183	C₂²⋊C₈	—	146	4	no	—
184	C₄×C₈	C₄×D₇₃	1	2	yes	—
185	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
186	C₂⁴	—	146	1	no	—
187	C₂×D₄	—	73	2	no	—
188	C₂×D₄	—	146	1	no	—
189	C₂×D₄	—	146	2	no	—
190	C₂×D₄	—	146	2	no	—
191	C₂²×C₄	—	292	2	no	—
192	C₂²×C₄	—	292	2	no	—
193	C₂²×C₄	—	292	2	no	—
194	C₂²×C₄	—	146	1	no	—
195	C₂²×C₄	—	146	2	no	—
196	C₂²×C₄	—	146	2	no	—
197	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
199	C₂²×C₄	#11924029	1	1	yes	characteristic
200	M₁₆	—	292	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	292	2	no	—
207	C₂×C₈	—	146	2	no	—
208	C₂×C₈	#11924005	1	2	yes	—
209	C₂×C₈	#11924005	1	2	yes	—
210	C₄⋊C₄	—	292	2	no	—
211	C₄⋊C₄	—	292	2	no	—
212	C₄⋊C₄	—	292	2	no	—
213	C₄⋊C₄	—	292	2	no	—
214	C₄⋊C₄	—	584	2	no	—
215	C₄⋊C₄	—	73	2	no	—
216	C₄⋊C₄	—	146	1	no	—
217	C₂²⋊C₄	—	146	2	no	—
218	C₂²⋊C₄	—	146	2	no	—
219	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
221	C₄²	Q₈×D₇₃	1	1	yes	characteristic
222	C₄²	#11924027	1	1	yes	characteristic
223	C₂³	—	292	2	no	—
224	C₂³	—	292	1	no	—
225	C₂³	—	292	1	no	—
226	C₂³	—	292	2	no	—
227	C₂³	—	146	2	no	—
228	C₂³	#14556566	1	1	yes	characteristic
229	D₄	—	292	2	no	—
230	D₄	—	146	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	292	2	no	—
233	C₂×C₄	—	292	4	no	—
234	C₂×C₄	—	292	2	no	—
235	C₂×C₄	—	292	2	no	—
236	C₂×C₄	—	584	2	no	—
237	C₂×C₄	—	584	2	no	—
238	C₂×C₄	—	584	2	no	—
239	C₂×C₄	—	146	2	no	—
240	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
241	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
242	C₂×C₄	#14556549	1	1	yes	characteristic
243	C₂×C₄	#14556562	1	1	yes	characteristic
244	C₂×C₄	#14556567	1	1	yes	characteristic
245	C₂×C₄	#14556571	1	1	yes	characteristic
246	C₂×C₄	#14556613	1	2	yes	—
247	C₂×C₄	#14556618	1	2	yes	—
248	C₂×C₄	#14556619	1	2	yes	—
249	C₂×C₄	#14556620	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	2	no	—
252	C₈	—	8	2	no	—
253	C₈	—	292	2	no	—
254	C₂²	—	292	2	no	—
255	C₂²	—	292	2	no	—
256	C₂²	—	292	2	no	—
257	C₂²	—	584	2	no	—
258	C₂²	—	584	2	no	—
259	C₂²	#17988444	1	1	yes	characteristic
260	C₂²	#17988803	1	2	yes	—
261	C₂²	#17988841	1	1	yes	characteristic
262	C₂²	#17988846	1	1	yes	characteristic
263	C₂²	#17988851	1	2	yes	—
264	C₄	—	2	1	no	—
265	C₄	—	2	1	no	—
266	C₄	—	292	2	no	—
267	C₄	—	1168	2	no	—
268	C₄	#17988584	1	1	yes	characteristic
269	C₄	#17988593	1	1	yes	characteristic
270	C₄	#17988626	1	2	yes	—
271	C₄	#17988830	1	2	yes	—
272	C₄	#17988888	1	2	yes	—
273	C₂	—	584	2	no	—
274	C₂	#22609493	1	2	yes	—
275	C₂	#22609507	1	2	yes	—
276	C₂	#22609942	1	1	yes	characteristic
277	C₂	#22609961	1	1	yes	characteristic
278	C₂	#22609987	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556566
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#334×C₇₃	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂²×D₂₉₂	D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#334	D₇₃
hypocenter	C₇₃	#8656
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #22610059	C₂	semidirect
4. #22607815	C₂	semidirect
5. C₂×#334×C₇₃	C₂	semidirect
9. #17988666	C₄	semidirect
143. C₂×#334	D₇₃	semidirect
145. C₇₃	#8656	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17988666	2
#22610059	2
#22609932	2
#22607815	2
C₂×#334×C₇₃	2
#8656	73